A novel approach to sheaves on diffeological spaces

Nezhad, Akbar Dehghan; Ahmadi, Alireza

doi:10.1016/j.topol.2019.05.006

Cited by 7 publications

(9 citation statements)

References 4 publications

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“…If d f x is an epimorphism at each point, then ψ • F is a diffeological submersion. Hence by Lemma 5.11 (5), f is locally a diffeological submersion and so f itself is a diffeological submersion. If d f x is a monomorphism at each point, then ψ • F is a diffeological immersion.…”

Section: Proposition 515 a Map Between Diffeological Spaces Is A Diff...mentioning

confidence: 87%

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Submersions, immersions, and étale maps in diffeology

Ahmadi¹

2022

Preprint

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One of the most important challenges in diffeology is providing suitable analogous for submersions, immersions, and étale maps (i.e., local diffeomorphisms) consistent with the classical versions of these maps between manifolds. In this paper, we introduce diffeological submersions, immersions, and étale maps as an adaptation of these maps in diffeology by a nonlinear approach. We explore their behaviors from different aspects in a systematized manner with respect to plots, and especially, show that for manifolds, there is no difference between the classical versions and diffeological versions of these maps. Moreover, we introduce a class of diffeological spaces, called diffeological étale manifolds, and establish the rank theorem for them. It is also proved that a diffeological étale space on a diffeological orbifold is again a diffeological orbifold.2010 Mathematics Subject Classification. 57P99, 57R99. Key words and phrases. diffeological spaces, submersions, immersions, embeddings, étale maps and spaces. 1 By an étale map, we mean a local diffeomorphism with respect to the D-topology (see also [8, §2.5]), and an étale space is the source of an étale map.

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Section: Proposition 515 a Map Between Diffeological Spaces Is A Diff...mentioning

confidence: 87%

“…We denote by Etale(X) the collection of diffeological étale spaces on a diffeological space X as a full subcategory of the comma category Diff/X, or equivalently, that of comma sheaves (see [5]). That is, a morphism φ between diffeological étale spaces E and E ′ on X is a commutative diagram…”

Section: éTale Maps In Diffeologymentioning

confidence: 99%

Submersions, immersions, and étale maps in diffeology

Ahmadi¹

2022

Preprint

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“…In [8], three versions of Čech cohomologies are suggested as those on the site of plots, the site of D-open subsets, and as the cohomology of the associated cochain complex of sections of a presheaf. In Appendix A, we show that the last two are the same.…”

Section: Introductionmentioning

confidence: 99%

“…We characterize the isomorphism classes of diffeological fiber, principal, and vector bundles as (non-abelian) diffeological Čech cohomology in degree 1. In Appendix A, we conclude the paper with a detailed discussion on the Čech cohomologies suggested in [8], as well as, a remark on the Čech cohomology due to Krepski et al…”

Section: Introductionmentioning

confidence: 99%

Diffeological Čech cohomology

Ahmadi¹

2023

Preprint

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Motivated by problems in which data are given over covering generating families, we suggest a new cohomology theory for diffeological spaces, called diffeological Čech cohomology, which is an exact ∂ -functor of the section functor for sheaves on diffeological spaces. As applications, under the situations of a setup, i) the generalized Mayer-Vietoris sequence for a diffeological space is established; ii) a version of the de Rham theorem is obtained, which connects diffeological Čech cohomology to the de Rham cohomology. Moreover, we characterize the isomorphism classes of diffeological fiber, principal, and vector bundles as (non-abelian) diffeological Čech cohomology in degree 1.

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“…Sheaves and cosheaves are robust tools to study local information on sites (categories with Grothendieck topologies), given by functors that preserve (co)limits over coverings. Sheaves and quasi-sheaves on diffeological spaces were introduced by the authors [9] with respect to the site of plots and covering generating families, respectively, to study relations between data on spaces and those on plots. In this paper, we investigate cosheaves on diffeological spaces, defined as cosheaves on the site of plots (Definition 3.2).…”

Section: Introductionmentioning

confidence: 99%

Some aspects of cosheaves on diffeological spaces

Ahmadi

Nezhad

2020

CGASA

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We define a notion of cosheaves on diffeological spaces by cosheaves on the site of plots. This provides a framework to describe diffeological objects such as internal tangent bundles, the Poincaré groupoids, and furthermore, homology theories such as cubic homology in diffeology by the language of cosheaves. We show that every cosheaf on a diffeological space induces a cosheaf in terms of the D-topological structure. We also study quasicosheaves, defined by pre-cosheaves which respect the colimit over covering generating families, and prove that cosheaves are quasi-cosheaves. Finally, a so-called quasi-Čech homology with values in pre-cosheaves is established for diffeological spaces.

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A novel approach to sheaves on diffeological spaces

Cited by 7 publications

References 4 publications

Submersions, immersions, and étale maps in diffeology

Submersions, immersions, and étale maps in diffeology

Diffeological Čech cohomology

Some aspects of cosheaves on diffeological spaces

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